A147972 Smallest prime p modulo which the first n primes are nonzero quadratic residues.

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 23616331489, 23616331489, 89206899239, 121560956039, 196265095009, 196265095009, 513928659191, 5528920734431, 8402847753431, 8402847753431, 8402847753431, 70864718555231

Offset: 1

Author

Max Alekseyev, Nov 18 2008

Keywords

• nonn

Comments

The same primes without repetitions are listed in A147970.

a(n) <= min{A002223(n), A002224(n)}. What is the smallest n for which this inequality is strict?

By definition, a(n) == 1, 7 (mod 8), so a(n) = min{A002223(n), A002224(n)}. - Jianing Song, Feb 18 2019

Crossrefs

Cf. A000229, A002189, A002223, A002224, A045535, A053760, A133435, A147969, A147970, A147971.

Smallest prime p such that each of the first n primes has q q-th roots mod p: this sequence (q=2), A002225 (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).

Programs

Mathematica

(*version 7.0*)m=1;P=7;Lst={p};While[m<25,m++;S=Prime[Range[m]];While[MemberQ[JacobiSymbol[S,p],-1],p=NextPrime[p]];Lst=Append[Lst,P]];Lst (* Emmanuel Vantieghem, Jan 31 2012 *)

PARI

t=2;forprime(p=2,1e9,forprime(q=2,t,if(kronecker(q,p)<1,next(2)));print1(p", ");t=nextprime(t+1);p--) \\ Charles R Greathouse IV, Jan 31 2012

Formula

a(n) >= min{A002189(n-1), A045535(n-1)}. - Jianing Song, Feb 18 2019

Extensions

a(23)-a(25) from Emmanuel Vantieghem, Jan 31 2012

a(26)-a(37) from Max Alekseyev, Aug 21 2015